Question

(3) (10 points) Let H and K be non-empty subsets of a vector space V. The sum of H and K, written H + K, is the set of all vectors in V that can be written as the sum of two vectors, one in H and the other in K: that is H + K = {W EVw = u + v, for some u E H and v EK}. Show that if H and K are subspaces of V, then H + K is a subspace of V.

Answer 1

coly subset let Vector The Hand k space sum of be non-empty v. H and k is defined as, H+K=swer w = u tū for some u thy Tek Suppose to that H and k are subspace of - claim: H+K is a subpace of V. Here sufficient to prove that, 2 w, tw2 E H t k t cos , WzEHK 1x wi tHtl + XEIF F f io, t Hik let , w2EHtk be any vectors 4 X Elf any scalass- To = uit W2 = a + 2 tu, i ųEHT - v, U EK. Consider wit We = - - a uitvi)+(uztve) m+ (Titu) th ut cuz tī, ) t ...' ... T is associali Tot i = situ = itun) + (v tv2) + 0, е н+с - uita EHP j , tuzek Cas Hrk are hebspace to, + We thak & w, Wethel

by using definition i was solved this problem

consider = x u tavi 1 xw, € Htk a u € HT QUICK (as HTK are subspace) xw, EH+K + XtIFF f witHtK From Htk is subspace of v